Question

I need Help with this question.

Please show workings.

Question:

`{3}^(x) + {4}^(x) = {5}^(x)`



P.S : The answer is 2.
I just need the working.​​

Answer 1

`x =2`

Explanation:

Given :-

• 
  `{3}^(x) + {4}^(x) = {5}^(x)`

And we need to find out the value of x. Well there is no specific method to solve the equation .This can be only done using the " Trial and error" Method.

• We know that , 3 , 4 and 5 are Pythagorean triplets . So the sum of squares of two smallest numbers is equal to the square of the largest number . Henceforth ,

`\implies {3}^(2) + {4}^(2) = {5}^(2)`

Verification :-

`\implies {3}^(2) + 4^2 = 9+16=25=\boxed{5^2}`

So , the value of x is 2 . We can here prove that , x does not have other roots other than 2 . For that , divide the both sides of equation by
`5^x` , we have ,

`\implies \frac{{3}^(x) + {4}^(x)}{5^x} = \frac{{5}^(x)}{5^x}`

`\implies \bigg( (3)/(5)\bigg)^x+ \bigg( (4)/(5)\bigg)^x = 1`

• Now if we take the value of x greater than 2 or less than 2 , then the value 1 will not be satisfied for the values of x greater than or less than 2 .

That is ,

• If x > 2

`\implies \bigg( (3)/(5)\bigg)^x+\bigg( (4)/(5)\bigg)^x > \bigg( (3)/(5)\bigg)^2 +\bigg( (4)/(5)\bigg)^2`

Subsequently :-

`\implies \bigg( (3)/(5)\bigg)^x+\bigg( (4)/(5)\bigg)^x > 1`

• If x < 2

`\implies \bigg( (3)/(5)\bigg)^x+\bigg( (4)/(5)\bigg)^x < \bigg( (3)/(5)\bigg)^2 +\bigg( (4)/(5)\bigg)^2`

Subsequently :-

`\implies \bigg( (3)/(5)\bigg)^x+\bigg( (4)/(5)\bigg)^x <1`

• Thus there is no other value other than 2 for which the value of above expression becomes 1 .

Hence 2 is the root of the given equation.